
Given a Zariski-open immersion \(j:U \rightarrow X\)  and a Zariski-constructible sheaf \({F}\)  on \(U\) , the extension \(j_! {F}\)  can fail to be Zariski-constructible \(X\) , unlike the situation in algebraic geometry (see next example). The main problem is that the operation of taking Zariski-closures in \(X\)  of Zariski-closed subsets of \(U\)  is poorly behaved in general (e.g., it does something non-trivial over \(U\) ); this issue does not arise if \({F}\)  is itself locally constant.
Example 3.3 Let \({F}\)  be the direct sum of skyscraper sheaves supported at an infinite discrete set of classical points in \((\mathbf {A}^1)^{an}\) , and let \(j:(\mathbf {A}^1)^{an} \rightarrow (\mathbf {P}^1)^{an}\)  be the standard open immersion. Then \(j_! {F}\)  is not Zariski-constructible on \((\mathbf {P}^1)^{an}\) : any Zariski-closed set of \((\mathbf {P}^1)^{an}\)  must be either finite or all of \((\mathbf {P}^1)^{an}\)  by rigid GAGA.

This phenomenon should not be regarded as a pathology: similar examples occur in complex analytic geometry as well, and are a natural consequence of the non-quasi-compactness of affine space in any kind of analytic geometry.

Huber's book [1]} defines a notion of “constructible” sheaves that is very well-behaved from a topos theoretic perspective. However, these sheaves are typically not Zariski-constructible; for instance, if \(j\)  is the qcqs open immersion defined by including a closed disc of radius (say) \(1/2\)  inside a closed disc of radius 1, then \(j_! \Lambda \)  is constructible in Huber's sense but is not Zariski-constructible. In fact, the overlap between these two notions is exactly the lisse sheaves.


